Research in physical applied mathematics, motivated by real world
problems is central to my investigations. My research is generally in
the field of nonlinear waves with a focus on two areas: 1) fluid
dynamics of dispersive media with accompanying wave excitations
including dispersive shock waves and solitary waves; 2) dynamics of
ferromagnetic media, spin torque, and localized excitations in
nanomagnetism. Methods employed include mathematical modeling,
analysis, asymptotics, Whitham modulation theory, and numerical
analysis. Whenever possible, comparisons with experiment are carried
out.
Current research involves the construction and stability of dispersive
shock waves in multiple dimensions with applications to boundary value
problems in supersonic Bose-Einstein condensate and nonlinear optical
flows. This research is supported by the National Science Foundation
through an applied math individual investigator grant DMS 1008973.
In the field of magnetodynamics, I am currently studying the
excitation of localized wave structures via the spin torque effect and
their coherent propagation.
A dynamic, visual representation of some of my scientific work is
exhibited below with animations of numerical simulations from some
problems that I have considered. Numerical simulations are excellent
for describing a particular problem with a particular set of
parameters. However, to understand the qualitative behavior, and to
be able to make informed predictions, mathematical analysis is
essential. The mathematical discussion has been kept to a minimum.
If you're interested in the math, read the papers! Not all of my
publications are represented here. To see a list,
click here. The animations can
be viewed with
Quicktime.
Note that some of the simulation files can be somewhat large.
Shock waves in classical, viscous fluids involve localized
"jumps" between different thermodynamic states. Shock waves in
dispersive fluids are no longer localized but still connect two
disparate fluid states with an expanding, oscillatory wavetrain.
Examples of such supersonic dispersive fluid dynamics have been
observed in superfluids such as Bose-Einstein condensates,
photonic "fluids" where intense light propagates through a
defocusing, nonlinear medium, shallow water waves, atmospheric
waves, and plasma. Dispersive fluids can often be described by
nonlinear wave equations. Well known (and well studied)
examples include the Korteweg-deVries equation (KdV) and the
Nonlinear Schrodinger equation (NLS).
click images to start movies
Supersonic, two-dimensional nonlinear Schrodinger flow from
left to right past a corner leads to the formation of oblique
DSWs. Depending on the Mach number of the upstream flow and the
corner angle, different dynamical instabilities are observed.
Upper left: convective instability. Upper right: absolute
instability. Lower left: convective instability with cavitation.
Lower right: absolute instability with cavitation.
The classical problem of a piston (not visible in the
animations) compressing a dispersive fluid is considered. A pure
DSW is excited for positive piston speeds below a critical
threshold (top animation). When the piston speed exceeds the
threshold (bottom animation), a DSW with an oscillatory wake and
saturated amplitude is excited. The upper (lower) panel in each
animation corresponds to the density (velocity) of the dispersive
fluid. The reason for the rapidly varying behavior of the
velocity at the left (inside the "piston") is due to the fact that
the fluid density is negligible there so that the velocity (the
gradient of the phase of a complex quantity) cannot be numerically
resolved.
The head on collision of two Nonlinear Schrodinger DSWs is
shown in the animation above. On the top (lower) left is the
fluid density (velocity). On the right is a phase diagram in
space-time predicting the number of phases required to describe
the asymptotic solution. The two DSWs collide, exhibit a
multi-phase interaction for some time and then emerge from the
interaction with altered magnitudes and speeds. The vertical
dashed lines at left correspond to the boundaries of the predicted
phase regions.
click images to start movies
A faster Nonlinear Schrodinger DSW overtakes a slower one
leading to their eventual merger with a small multi-phase
interaction region.
By pulsing a repulsive laser through the center of a confined
BEC, circular rings of high density are formed. Nonlinear
self-steepening leads to the formation of
a dispersive
shock wave. The animation on the left results from laser
pulsing after the trap has been turned off. A careful analysis
shows that two DSWs are formed which eventually interact and
merge. This led me to consider DSW
interactions. The animation on the right corresponds to laser
pulsing while the BEC is in trap and results in the excitation of
one DSW.
Superfluid Dynamics in Bose-Einstein Condensates (BECs)
The interplay of cutting edge experiments and applied
mathematics is on display in the works shown below. A
Bose-Einstein condensate is a dilute gas of bosons (particles
with integer spin) that is carefully cooled to an extremely low
temperature resulting in remarkable quantum mechanical behavior
at the macroscopic scale. All simulations are performed in three
spatial dimensions corresponding to experimental conditions.
The animations render the vertically integrated BEC superfluid
density. The awesome experimentalists who actually make BECs
with whom most of this work is done
is Peter
Engels and his group in the Physics Department at Washington
State University.
click image to start movie
Sufficiently fast counterflow of two overlapping, nonlinearly
coupled BECs in a cigar shaped trap leads to modulational instability
and the formation of a number of beating dark-dark solitons. They
subsequently decay via a transverse instability into turbulent-like
behavior. Even though the image shows two physically separated
clouds, the two BECs are actually overlapping one another in space.
The counterflow of two overlapping, nonlinearly coupled BECs
in a cigar shaped trap leads to the formation of a dark-bright soliton
train and subsequent soliton interactions. Even though the image
shows two physically separated clouds, the two BECs are actually
overlapping one another in space. Counterflow induced modulational
instability is key to soliton train formation.
The classic problem from quantum mechanics
of matter
wave interference (see
the double
slit experiment). The macroscopic, nonlinear quantum nature
of BECs is on display with the interference of two BECs initially
separated by a repulsive laser barrier. When allowed to collide,
a regular fringe pattern is born that consists of a train of dark
solitons. We show that this interference pattern is made up of
two dispersive
shock waves placed back-to-back. While the dynamics here are
essentially one-dimensional, see BEC
merging for three-dimensional effects that can occur.
As in the matter wave
interference problem, a trapped, single-component BEC is
initially held separated into two pieces by a repulsive laser
barrier. When the barrier is removed, the two pieces combine
leading to the formation of essentially one-dimensional dark
solitons. In contrast to the interference problem, these solitons
rapidly decay into vortices via a transverse, snake instability
resulting in a prominent density bulge in the center of the cloud.
click image to start movie
This is the same simulation as above but zoomed in to reveal
the initially one-dimensional behavior and its subsequent breakup
into rich, complicated shapes.
Faraday waves are well known in viscous fluids (see,
e.g. Faraday
waves on cornflour). In the simulation above, you see the
formation of Faraday waves in a superfluidic BEC with no viscosity.
These standing waves are formed by periodically, radially "squeezing"
the cigar shaped trap that confines the BEC.
Ferromagnets consist of a lattice
of dipoles
that can precess much like a spinning top. However, unlike
spinning tops, the dipoles in a ferromagnet interact with one
another which can lead to the formation of waves. The work
described below focuses on the magnetic waves generated by the
spin torque effect whereby a DC current flowing through a magnet
(or a magnetic multilayer) can drive the magnetization (a
continuum approximation of the dipole lattice) away from
equilibrium. This leads to the generation of waves. The
magnetization (a 3-component vector) is visualized in several
ways.
click image to start movie
When a DC current flows through a nanocontact (the black
circle), a droplet soliton can be excited. This animation depicts
the "birth" of a droplet when the current is first turned on. An
instability leads to the initial generation of spin waves that
propagate away from the nanocontact and the eventual reversal of
the magnetization (red region). Once the droplet has formed, it
just "spins" in place, locally under the nanocontact. Contrast
this behavior of a ferromagnet with easy-axis anisotropy to that
of a ferromagnet with easy-plane
anisotropy. The frequency of spinning is in the gigahertz
(microwave) regime. The cones represent the orientation of the
local magnetization and the color corresponds to the component of
the magnetization in the perpendicular direction.
click image to start movie
In certain operating conditions, the droplet soliton can
experience a drift instability. The above animation shows this
effect. The droplet can move coherently! Eventually, its
amplitude decays due to magnetic damping.
A nanocontact abutting a single layer ferromagnet can excite
a variety of coherent, nonlinear wave structures including a
standing wave (top left), an oscillating vortex (top right), and a
collimated spin wave beam (bottom). The color corresponds to the
y-component of the magnetization and the arrows correspond
to the in-plane, x,y components.
This animation represents another way to visualize the
magnetodynamics of a spin torque driven nanocontact (not shown).
The amplitude and color correspond to one of the in-plane
components (the y-component) of the magnetization. In
contrast to the droplet soliton
excited in an easy-axis ferromagnet, here the excitation radiates
spin waves away from the nanocontact.
